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User blog:Heg Oiu/An exact definition of level difficulty
An exact definition of difficulty A definition of difficulty Here is an inductive definition of an optimal move and the probability of success of a board: * An optimal move is a move that maximizes the average probability of success of the possible resulting boards, assuming subsequent optimal moves; the probability of success of a board is the probability of success of an optimal move. * The probability of success with 0 moves left is 0 if you failed and 1 if you succeeded. Let the probability of success for a level be the average probability of success of all possible boards (with optimal moves). Let the hardness or difficulty of a level be the number of tries that is needed to succeed on the level with a probability of at least 0.5 (with optimal moves). Let the number of tries required to virtually guarantee success on a level be the number of tries needed to have at least a 0.95 chance of succeeding on the level. Then the difficulty and the "guarantee" numbers are ln(0.5)/ln(1-p) and ln(0.05)/ln(1-p) respectively, where p is the probability of success. The guarantee number is ln(0.05)/ln(0.5) ≈ 4.3 times larger than the difficulty number. Let the ''difficulty of an episode ''be the average difficulty of the levels (i.e. the sum of difficulties of the levels, divided by the number of levels of the episode). A categorization of levels and episodes Here is a categorization of the difficulty, d, of a level or an episode: *extremely easy, if d \leq 2^0 *very easy, 2^0 < d \leq 2^1 *easy, 2^1 < d \leq 2^2 *somewhat easy, 2^2 < d \leq 2^3 *somewhat hard, 2^3 < d \leq 2^4 *hard, 2^4 < d \leq 2^5 *very hard, 2^5 < d \leq 2^6 *extremely hard, 2^6 < d \leq 2^7 and, in general, the difficulty, d, of a level, or an episode, is in category c if 2^{c-1} < d \leq 2^c . How hard is Candy Crush? The difficulties of all current Candy Crush levels (without boosters) are at most very hard, i.e. ≤ 64, since, arguably, you are likely to succeed on any current level in at most 64 tries with very deliberate play. Some evidence for the above claim that all levels are at most very hard is needed. However aren't there videos available fairly quickly after release, from roughly the same players, of all levels (without boosters) that at least indicate this? And those players play fast, and sometimes not so deliberate or optimally. Anyway, FWIW, my experience is that there has been no level that required more than 64 tries with very deliberate play, so far. And very deliberate play is still worse than optimal play, especially on some levels. For an illustration of this, consider the challenge of getting 122 sugar drops on level 1476 (without boosters), which ought to be the most you realistically can get on a sugar drop level. With fast play, chances are that you will virtually never get it. With very deliverate play --- calculating the consequences of all promising moves, and then trying to figure out which is best --- you are likely to get the 122 drops in less than maybe 50 tries. And that very deliberate play still ought to be some distance away from optimal play. Category:Blog posts